We view a bisimplicial set \(X\)  as a simplicial diagram \(m \mapsto X_{m,\bullet }\)  in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\) .
Let \(J \in {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  denote the nerve of the groupoid with two objects and a unique isomorphism between them.
The following definitions are taken from [1]}, [2]}, [3]}.
